Sen. Bob Corker said a growing number of Republicans realize they can get the upper hand in budget negotiations with Democrats if they agree to raise tax rates on the wealthiest Americans.

In an interview on Fox News Sunday, the Tennessee Republican said he was inclined to support the tax rate increase on wealthy Americans, arguing it would raise about $400 billion to $800 billion in revenues, less than the tax increases Democrats have previously sought.

“There is a growing group of folks who are looking at this and realizing we don’t have a lot of cards as it relates to the tax issue before year’s end,” Sen. Corker said. “So a lot of people are putting forth a theory, and I think it has merit, where you go ahead and give the president … the rate increase on the top 2%, and all of a sudden the shift goes back to entitlements.”

Sen. Corker said that if the GOP grants Mr. Obama the tax increase, they can then use the looming vote on raising the debt ceiling to force cuts to entitlement programs. Mr. Corker did not specify what entitlements he was referring to. But Republicans have been talking about cuts to Medicare, including raising the age when retirees can enroll in the health care program. Others have been talking about adjustments to Social Security to curb costs of that program.

Mr. Corker said wealthy Americans already bear an unduly large tax burden, and over the long term he support tax reform to make the system more balanced. But in the short term agreeing to the Democratic proposal on raising tax rates will allow the Republicans to shift the debate to cuts in entitlement programs.

“Republicans know that they have the debt ceiling coming up right around the corner,” Sen. Corker said. “The leverage is going to shift to our side. We are going to do the same thing we did last time; if the president wants to raise the level by two trillion, we get two trillion in spending reductions.”

Democrats have pushed for raising the debt limit as part of the debate over averting the so-called fiscal cliff, the combination of tax increases and spending cuts scheduled to go into effect in January.

Mr. Corker said Republican House Speaker John Boehner would only agree to raising the debt limit if Democrats agree to cut entitlement spending.

“The only what the debt ceiling is given up is that the president comes to the table, talks to Speaker Boehner about real entitlement reform,” Sen. Corker said. “Without that there is no way the debt ceiling is going to be given up.”

Rep. Tom Cole (R., Okla.) delivered a similar message Sunday in an appearance on CNN’s State of the Union.

Mr. Cole said his constituents understand that their taxes are going up automatically early in the new year unless Congress acts, “and they would like to be taken out of the line of fire,” he said.

“They expect me to continue to fight for everybody’s taxes not going up,” he said. “But if I can get a deal that protects 98% of them and leaves me free to continue fighting for them, they would say, ‘Take that deal, that’s progress, that’s maybe working together across the aisle a little bit, and get it done.’”

In late November, Mr. Cole privately urged his Republican colleagues to extend middle-class tax cuts. On Sunday, he said Republicans would gain leverage if the issue was taken off the table, because Democrats would not be able to blame the GOP for increasing middle-class tax rates.

Despite the comments by Sen. Corker and others, the House leadership, at least publicly, remains staunchly opposed to higher rates and willing only to embrace increased tax revenues by eliminating or reducing tax deductions and other so-called loopholes. On NBC’s Meet the Press, Rep. Kevin McCarthy (R., Calif.), the House Majority Whip, said raising rates “doesn’t solve the problem.”

“If the president is asking for higher rates, he is asking for more revenue,” Rep. McCarthy said. “Most economists the best way to get that [revenue] is by closing special loopholes. When you close those it makes for a fairer tax process.”

Mr. Cole’s arguments were also met with skepticism by a GOP House colleague, Rep. Marsha Blackburn (R., Tenn.), who also spoke on CNN.

“I’m not sure there’s support for the rate hikes, there is support for revenue by cleaning up the [tax] code,” she said.

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Washington Wire is one of the oldest standing features in American journalism. Since the Wire launched on Sept. 20, 1940, the Journal has offered readers an informal look at the capital. Now online, the Wire provides a succession of glimpses at what’s happening behind hot stories and warnings of what to watch for in the days ahead. The Wire is led by Reid J. Epstein, with contributions from the rest of the bureau. Washington Wire now also includes Think Tank, our home for outside analysis from policy and political thinkers.